False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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False: every Henstock–Kurzweil integrable function is a derivative

Statement

False claim: Every Henstock–Kurzweil integrable function on an interval is the derivative of some function.

Facts & Assumptions

Given: The indicator $d$ of the irrationals on $[0, 1]$ .

[L1]

The indicator of the irrationals is Henstock–Kurzweil integrable with integral $1$ (The indicator of the irrationals is Henstock–Kurzweil integrable with integral $1$ ).

[L2]

Every derivative has the intermediate-value property (Darboux's theorem: every derivative has the intermediate-value property).

[L3]

The rationals and irrationals are both dense in $R$ (Both $Q$ and $R ∖ Q$ are dense in $R$ , and every nonempty open subset of $R$ is uncountable).

Refutation

technique · contradiction

1.1

By [L1], $d$ is HK integrable, and [L3] makes it take both values $0$ and $1$ on every nondegenerate subinterval while taking no value strictly between them.

givenL1L3

2.1

Suppose $d$ were a derivative; step 1.1 contradicts the intermediate-value property [L2], so the false claim is refuted.

step 1.1L2assume-contradischarge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

29 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Alessandro Fonda, The Kurzweil-Henstock Integral for Undergraduates, Ch. 1 (standard reference, not scraped)
Andrew Bruckner, Judith Bruckner and Brian Thomson, Real Analysis, Exercise 1:21.2 (standard reference, not scraped)